[Paper Review] Some classifications of biharmonic hypersurfaces with constant scalar curvature

Shun Maeta, Ye‐Lin Ou|arXiv (Cornell University)|Aug 28, 2017

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This paper classifies biharmonic hypersurfaces with constant scalar curvature in Einstein manifolds and space forms, proving that under certain curvature and integrability conditions—such as $ L^p $-bounded $ | abla H| $ or Ricci curvature bounds—complete biharmonic hypersurfaces are minimal or have constant mean curvature. Key results support Chen’s conjecture and the Balmuş-Montaldo-Oniciuc conjecture in specific geometric settings, particularly in spheres and non-positively curved Einstein spaces.

ABSTRACT

We give some classifications of biharmonic hypersurfaces with constant scalar curvature. These include biharmonic Einstein hypersurfaces in space forms, compact biharmonic hypersurfaces with constant scalar curvature in a sphere, and some complete biharmonic hypersurfaces of constant scalar curvature in space forms and in a non-positively curved Einstein space. Our results provide additional cases (Theorem 2.3 and Proposition 2.8) that supports the conjecture that a biharmonic submanifold in a sphere has constant mean curvature, and two more cases that support Chen's conjecture on biharmonic hypersurfaces (Corollaries 2.2,2.7).

Motivation & Objective

• To classify biharmonic hypersurfaces with constant scalar curvature in Einstein manifolds and space forms.
• To provide further evidence for Chen’s conjecture that biharmonic submanifolds in Euclidean space are minimal.
• To support the Balmuş-Montaldo-Oniciuc conjecture that biharmonic submanifolds in spheres have constant mean curvature.
• To establish conditions under which complete biharmonic hypersurfaces with constant scalar curvature are minimal or have constant mean curvature.
• To extend results on biharmonic hypersurfaces in non-positively curved Einstein spaces using geometric analysis and maximum principles.

Proposed method

• Uses the biharmonic equation for hypersurfaces in Einstein manifolds, derived from the critical point condition of the bi-energy functional.
• Applies the Gauss and Weingarten equations to relate curvature invariants of the hypersurface and ambient space.
• Employs the Bochner formula for $ | abla H|^2 $, combining Ricci curvature bounds and the biharmonic equations to derive differential inequalities.
• Applies maximum principles (ii) and (iii) from Theorem 2.5 to $ | abla H|^2 $, showing it is constant under $ L^p $ or curvature-bounded conditions.
• Uses Newton’s inequality $ |A|^2 angle mH^2 $ to estimate the Laplacian of $ | abla H|^2 $, leading to $ \Delta|\nabla H|^2 \geq \varepsilon|\nabla H|^2 $ under curvature assumptions.
• Analyzes the structure of the mean curvature function $ H $ via the first biharmonic equation $ \Delta H - H(|A|^2 - \lambda) = 0 $, concluding $ H = 0 $ if $ |A|^2 = \lambda \leq 0 $.

Experimental results

Research questions

• RQ1 Under what conditions is a complete biharmonic hypersurface with constant scalar curvature in a non-positively curved Einstein manifold minimal?
• RQ2 Does a complete biharmonic hypersurface in a sphere with constant scalar curvature and $ H^2 \geq \frac{2\varepsilon + 4}{m(5m + 4)} $ have constant mean curvature?
• RQ3 Can $ L^p $-integrability of $ |\nabla H| $ or Ricci curvature bounds imply constancy of $ H $ or minimality?
• RQ4 How do curvature bounds and the Bochner formula constrain the mean curvature function $ H $ in biharmonic hypersurfaces?
• RQ5 To what extent do the results support Chen’s conjecture and the Balmuş-Montaldo-Oniciuc conjecture in the context of constant scalar curvature?

Key findings

• A complete biharmonic hypersurface with constant scalar curvature in a non-positively curved Einstein manifold is minimal if $ |\nabla H| \in L^p $ for some $ 2 < p < \infty $, or if $ |\nabla H| \in L^2 $ and Ricci curvature is bounded below by $ -c(1 + r^2(x)) $.
• If $ H^2 \geq \frac{2\varepsilon + 4}{m(5m + 4)} $ for some $ \varepsilon > 0 $, then a complete biharmonic hypersurface in $ S^{m+1} $ has constant mean curvature under conditions (A), (B), or (C) on $ |\nabla H| $ and Ricci curvature.
• Corollary 2.7 confirms a partial case of Chen’s conjecture: any complete biharmonic hypersurface with constant scalar curvature and $ |\nabla H| \in L^p $ for $ 2 < p < \infty $ in Euclidean space is minimal.
• Theorem 2.3 provides a new case supporting the Balmuş-Montaldo-Oniciuc conjecture, showing that biharmonic hypersurfaces in $ S^{m+1} $ with constant scalar curvature and sufficient $ H^2 $-lower bound have constant mean curvature.
• Proposition 2.6 establishes that in non-positively curved Einstein spaces, such hypersurfaces are minimal under $ L^p $ or curvature-bounded $ |\nabla H| $, due to the non-positivity of $ \lambda $ and $ |A|^2 = \lambda \leq 0 $ implying $ |A|^2 = 0 $.
• The analysis shows $ \Delta|\nabla H|^2 \geq \varepsilon|\nabla H|^2 $ under the given assumptions, leading to $ |\nabla H|^2 $ being constant and ultimately $ H $ constant or zero via the biharmonic equations.